Perhaps it might be a good idea to find out what Riemann's
formula is? Then try comparing your Excel power-law fits
with Gram's convenient version of Riemann's formula, which
you will find in Eq (12) at

Perhaps it might be a good idea to find out what Riemann's
formula is? Then try comparing your Excel power-law fits
with Gram's convenient version of Riemann's formula, which
you will find in Eq (12) at

> > What if Riemann's prime-counting formula was not the best?
>
> Perhaps it might be a good idea to find out what Riemann's
> formula is? Then try comparing your Excel power-law fits
> with Gram's convenient version of Riemann's formula, which
> you will find in Eq (12) at

Perhaps it might be a good idea to find out what Riemann's
formula is? Then try comparing your Excel power-law fits
with Gram's convenient version of Riemann's formula, which
you will find in Eq (12) at

Given various exact formulas and algorithms for computing pi(x),
the "best" would seem to be the one permitting the fastest computation (maybe memory
consumption should also be taken into account).
And in fact, the fastest currently known algorithm (asymptotically) for computing
pi(x) is based on a form of Riemann, not combinatorial counting methods and not just generating primes. Also, for inexact computation of pi(x) to specified accuracy, this method still seems best known.

However, there is no reason to believe that further speedups are necessarily impossible
with further algorithmic/analytic ideas nobody has thought of yet.
It would help if anybody had the faintest idea how to prove lower bounds on the
computational complexity of counting primes.

I could continue. With Green-Tao, we know there are arbitrarily long arithmetic progressions with all-prime entries. How computationally complex is it to find an N-term example? Well... it seems to be extremely hard... quite likely humanity will
never find an example with N=50... but I am unaware of any
reasonable lower bounds, or upper bounds, on this computational complexity!

Chris De Corte

Alan, I said in my document that I only made a calculation up to 49978001 because I don t have other data available. If you all agree on a new trial then

Message 10 of 16
, Jul 28, 2013

Alan,

I said in my document that I only made a calculation up to 49978001 because I don't have other data available.

If you all agree on a new trial then please provide me with 10 N, 10 pi(x) and 10 best Riemann's approximations and I will try to calculate a new alpha and beta.